<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\chapter{SPECIFIC HEAT CAPACITY OF METALS \label{ch.SpHtKRN}}
\bigskip
\subsection*{Objectives}
%\bigskip
\noindent
%P\aa\ slutten av 1800-tallet og begynnelsen av 1900-tallet, kom det til syne en rekke eksperimentelle 
The principal objective is to determine the specific heat capacity of given metallic specimens. Some other obejectives are:---
\begin{itemize}
  \item To grasp the physical concepts and mathematical relations concerning the gain and loss of heat by objects
  \item To appreciate the role played by measurements of heat capacity in the development of quantum theory
  \item To learn to fit experimental data to theoretical expressions, to acquire the ability to change to from one system of units to another, and to become aware of the value as well as the limitations of a scientific model.
\end{itemize}
\section{Theoretical background}       % Enter section title between curly braces

\subsection{Introduction to calorimetry}    % Enter subsection title between curly braces

The specific heat capacity (or simply the specific heat) of a substance, usually denoted by $c$, is defined as the quantity of heat required to raise the temperature of a unit mass of the substance through 1 $^\circ$C (or 1 K); its value depends on the initial temperature itself. Thus if a quantity of heat   raises the temperature of a body (of mass $m$) from  $T_1$ to $T_2$, the mean specific heat $\overline{c}$  is given by $Q/m(T_2-T_1)$ ; if the temperature increases by a small amount, from $T$  to $T+\Delta T$ , the specific heat at the temperature $T$ is ($1/m)(\Delta Q/\Delta T)$. The thermal capacity of a body is equal to the product of its mass and specific heat.

You will be following what is known as the method of mixtures. It consists in transferring the quantity of heat that is to be measured to a known mass of water in an insulated vessel (called a calorimeter) of known thermal capacity and measuring the resulting change in temperature. Suppose that a substance of mass  $m_1$, specific heat $c_1$  and initial temperature $T_i$  is plunged into a body of water (with mass $m_2$ and specific heat $c_2$) and temperature  $T^\prime$, and let $W$  and $T_f$  denote, respectively, the thermal capacity of the calorimeter and the final temperature of the mixture. Upon equating the heat lost by the substance to the heat acquired by the calorimeter and water, we get
\begin{equation}\label{eq:heatbalance}
m_1c_1(T_i-T_f)=(m_2c_2+W)(T_f-T_i),
\end{equation}
so that
\begin{equation}\label{eq:cOne}
c_1 =\frac{(m_2c_2+W)(T_f-T_i)}{m_1(T_i-T_f)},
\end{equation}
The above relation provides the basis for the determination of the specific heat, but for precise measurements, various corrections are needed to compensate for the heat lost by the system to the surroundings. See Appendix I for further details.

Sometimes it is more convenient to express specific heat in terms of the molar heat capacity, which is simply the energy required to raise the temperature of one mole of the material by one degree Celsius or one Kelvin. It is usually denoted by the symbol $C$, and one adds a subscript ($p$  or   $v$ so as be able to specify the parameter, namely pressure or volume, that is held fixed when the temperature is allowed to change).

\subsection{Classical view of specific heats (based on kinetic theory)} 

If one regards the atoms of a monoatomic gas as structureless point masses, each with three degrees of freedom, one mole of the substance will have altogether $3N$  degrees of freedom, where $N$ is the number of molecules in one mole of the gas.  In the case of a perfect gas, the molecules will have no potential energy, and the total energy of a mole of a monoatomic gas will therefore be
\begin{equation}\label{eq:ClassicalE}
E=3N\times \textstyle{\frac{1}{2}}k_{\rm B}T=\frac{3}{2}RT
\end{equation}                                                  
which leads one to the following expression for the molar specific heat:
\begin{equation}\label{eq:ClassicalCv}
C_v=\frac{\partial E}{\partial T}=\textstyle{\frac{3}{2}}R.
\end{equation}    	                                                               
For all perfect gases  $C_p-C_v=R$, which implies that
\begin{equation}\label{eq:ClassicalCv}
C_p=\textstyle{\frac{5}{2}}R.
\end{equation}    	                                                               
The above predictions are in excellent agreement with the behaviour of noble gases.

In a classical description, the molecules of a diatomic gas are viewed as two atoms joined {\it rigidly\/} to one another. Treating the atoms as points, each molecule will have three degrees of freedom of translation and two of rotation about the two axes which are perpendicular to the interatomic axis; the argument for ignoring one rotational degree of freedom was itself rather shaky, but we will not go into this detail here. Thus the total energy of one mole of such a gas will be  $E=\textstyle{\frac{5}{2}}RT$, and
\begin{equation}\label{eq:ClassicalC2at}
C_v=\textstyle{\frac{5}{2}}R,\quad  C_p=\textstyle{\frac{7}{2}}R.
\end{equation}  
This is approximately true for some gases at around 300 K. The main problem here is that the specific heats are predicted to be independent of temperature, and this prediction is not borne out by experimental observations (see below).
	An expression for the molar specific heats of solids may be obtained by following the same reasoning. A crystal composed of $N$ atoms will have   $3N$ modes of vibration, and there will be $6N$ degrees of freedom in all, each with an energy content of $\frac{1}{2}k_{\rm B}T$; the molar heat capacity will therefore be $C_v=3R$, a relation known as the law of Dulong and Petit. However, measurements show that the specific heat of a solid depends on its temperature, and the rate of decline is very steep at low temperatures. Even at room temperature, the molar heat capacities of some substances, including diamond (the example used by Einstein), are significantly smaller than $3R$.

It had become obvious, long before the end of the nineteenth century, that the variation of specific heats of gases and solids could not be explained by the theories which had otherwise proved to be highly successful. The first important step towards a satisfactory explanation of the phenomenon, at least in the case of solids, was taken by Einstein.
\subsection{Einstein’s theory of the heat capacity of a solid}  

Einstein assumed that a crystal comprising of $N$ atoms can be treated as an assembly of $3N$ distinguishable one-dimensional oscillators, and added the requirement that the energy of each oscillator is quantized according to the following relation
\begin{equation}\label{eq:Quantize}
E=kh\nu = kE_1=E_k,\quad k=0,1,2,\dots
\end{equation}  %\frac{}{} 
The average energy of a single oscillator comes out to be
\begin{equation}\label{eq:AvEn}
\langle E \rangle =\frac{E_1\e^{-E_1/k_{\rm B}T}+E_2\e^{-E_2/k_{\rm B}T}+\ldots}{1+\e^{-E_1/k_{\rm B}T}+E_1\e^{-E_2/k_{\rm B}T}+\ldots}=
\frac{E_1\e^{-E_1/k_{\rm B}T}}{1-\e^{-E_1/k_{\rm B}T}}=\frac{E_1}{\e^{E_1/k_{\rm B}T}-1}.
\end{equation} 
Whence follows the result derived by Einstein:
\begin{equation}
C_v=3R\left ( \frac{E_1}{k_{\rm B}T} \right )^2\frac{\e^{-E_1/k_{\rm B}T}}{[1-\e^{-E_1/k_{\rm B}T}]^2} =
    3R\left ( \frac{E_1}{k_{\rm B}T} \right )^2\frac{\e^{E_1/k_{\rm B}T}}{[\e^{E_1/k_{\rm B}T}-1]^2}.
\label{eq:EinCv}
\end{equation} 
You should convince yourself that $C_v$  tends to zero as the temperature approaches zero, and that the law of Dulong and Petit is recovered at high temperatures. It is customary to make the substitution $\Theta_{\rm E}=E_1/k_{\rm B}$, and refer to $\Theta_{\rm E}$  as the Einstein temperature of the solid under consideration.

\section{Pre-lab tasks}   

You are supposed to complete the following tasks before starting the experiment. You should show your work to, and secure the approval of, the person in charge of the laboratory work. 

\subsection{Diamond: Einstein's choice}

Einstein, who used the symbol $c$ instead of our $C_v$, used the data for diamond for testing his theory.  His own figure (from p. 186 of his article) and the entries in his table (p. 190) are reproduced below.



\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/EinsteinHeer.TIF
  \includegraphics[bb=0 0 437 271,width=10cm,height=6.2cm,keepaspectratio]{EinsteinHeer.tif}
  \caption{\small Einstein plotted the heat capacity of diamond (circles) as a function of temperature to show that atomic vibrations in solids are quantized. The temperature (x axis) is $T/\Theta_{\rm E}$. The heat capacity (y axis) is given in cal/(mol K). The smooth curve is his prediction.}
  \label{fig:EinsteinHeer}
\end{figure}

\bigskip

\centerline{\small Table 1: Einstein's table} 
\centerline{\includegraphics[bb=0 0 249 296,width=6cm,height=7.13cm,keepaspectratio]{EinsteinTable.tif}}



You will notice that the axes in his plot are not labelled, an omission that will not be tolerated in our times. To make sense of his plot, note first that heat was measured, in Einstein’s time, and even until much later, in calories. Find out the relation between a calorie and a Joule, the modern unit, since you are going to need it later. The horizontal axis is in fact $x$, which equals, in our notation, the dimensionless quantity $T/\Theta_{\rm E}$. {\it One entry in the third column of Einstein’s table is clearly wrong, and it is your task to spot and correct the typographic error\/}.

Plot the experimental values of $C_v$ against $T$. It is a simple matter to infer, from Einstein’s table, his choice for $\Theta_{\rm E}$. Let us pretend, however, that the third column of his table is not available to us. Insert a first guess for  $\Theta_{\rm E}$ in Eq.~(\ref{eq:EinCv}) and see how well the calculated values agree with the experimental points. By trial and error you should soon be able to get values that seem, on a visual examination, to be close to those falling on the smooth dashed curve in Figure 1. If you already know a better way of using Excel for data fitting, you are free to use that, but for this particular problem you need nothing more than a few rounds of trial and error. A result obtained by KRN is reproduced in Figure 2 to serve as a guide. Notice that the theoretical curve can be calculated for any value of $T$, and a smooth curve can only be obtained by choosing a relatively dense set of temperatures, which is not a hardship if one is using a computer.

%\medskip

\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Figure2.TIF
  \includegraphics[bb=0 0 605 441,width=10cm,height=7.29cm,keepaspectratio]{Figure2}
  \caption{\small The specific heat data in Table 1 have been plotted against the temperature $T$. By choosing a suitable value for$\Theta_{\rm E}$ you should be able to reproduce the smooth theoretical curve. }
  \label{fig:Figure2}
\end{figure}


%\centerline{\includegraphics[bb=0 0 605 441,width=9cm,height=6.56cm,keepaspectratio]{Figure2.tif}}




\subsection{Data on aluminium}
We will now consider aluminium (symbol: Al). Let us begin by using the data reported by Giuaque and Meads, who measured the molar heat capacity of Al from 15 to 300 K. They chose the units Cal.deg$^{-1}$g.-atom$^{-1}$. If you have not come across the obsolete terms gram-atom and gram-molecule, it is time to find out by consulting, for example, the {\it Wikipedia\/}. For your convenience, their data have been tabulated in Table 2. Convert these values to modern units, namely J $^\circ$C$^{-1}$ mol$^{-1}$ or J K$^{-1}$ mol$^{-1}$, and repeat the steps taken in task 1 above. Using visual judgment you should end up with a plot like that appearing in Figure 3.

%Let us recall, for possible later use, the opening sentence of the paper by Giauque and Meads: ‘The heat capacities of gases can often be calculated from molecular data more accurately than they can be obtained by direct measurement but this has not yet been found possible for even the simplest crystalline solids’. By the term `molecular data', they mean spectroscopic data.


\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Figure3.TIF
  \includegraphics[bb=0 0 656 440,width=10cm,height=6.71cm,keepaspectratio]{Figure3}
  \caption{\small A plot of the data in Table 2; other comments as in Figure 2. }
  \label{fig:Figure3}
\end{figure}


%\centerline{\includegraphics[bb=0 0 656 440,width=9cm,height=6.04cm,keepaspectratio]{Figure3.tif}}

A more extensive data set for the molar heat capacity of Al was made available by Buyco and Davis.  A selection of their entries, covering the temperature range 10--900 K, is tabulated. The authors have used a different system of units by stating their values in cal/(g.K).  For your convenience, some of their data have been gathered in Table 3 and plotted in Figure 4. Convert these entries to SI units, and repeat the steps taken in task 1 above. The discrepancy between the experimental values and Einstein’s predictions for larger temperatures cannot be repaired by a different choice of the Einstein temperature. Can you think of an explanation? 
  
%\centerline{\includegraphics[bb=0 0 656 440,width=9cm,height=6.04cm,keepaspectratio]{Figure4.tif}}

\begin{figure}[tbp] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Figure4.TIF
  \includegraphics[bb=0 0 656 440,width=10cm,height=6.71cm,keepaspectratio]{Figure4}
  \caption{\small A plot of the data in Table 3.}
  \label{fig:Figure4}
\end{figure}


\vspace{-5mm}

\subsection{Specific heat of a diatomic gas}
Einstein's picture of a solid is very close (but not identical) with our current visualization of an ideal gas of diatomic molecules. We would expect that a  $C_v$-versus-$T$ plot for such a gas will resemble the corresponding curve for a solid. Table 4 gives experimental values of $C_p$ for F$_2$. Convert these values to $C_v$  and plot the resulting values against $T$. Compare your plot with Figure 4 and comment on the similarities and differences.

{\small {\it Optional\/}. From an analysis of the vibrational spectrum of F$_2$ one finds that, in the zero-order simple-harmonic approximation, the energy (in spectroscopic units) of a vibrational quantum of a molecule of F$_2$ is  711 cm$^{-1}$. Use this information to predict the values of $C_v$, and compare the predicted values with the experimental results.}

\subsection{The actual experiment}
%\setcounter{figure}{4}
The experiment itself is rather simple. The major item of equipment is an electronic balance, depicted in Figure 5. A metallic specimen, initially close to room temperature, is immersed in liquid nitrogen, and the amount of nitrogen which boils off is measured using a weighing balance. This quantity is then used in a heat balance equation to determine, in the manner described below, the change in the internal energy of the metal and the Einstein temperature.

\vspace{-5mm}
\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Balance.PNG
  \includegraphics[width=8.17cm,height=10cm,keepaspectratio]{Balance}
\vspace{-10mm}
  \caption{\small A view of the electronic balance to be used in your experiment.}
  \label{fig:Balance}
\end{figure}

\vspace{5mm}


An insulated container is made by placing one polystyrene coffee cup inside the other. Some liquid nitrogen (about 25 g) is poured into the container, which is then placed on the balance together with the metallic specimen whose heat capacity is to be measured. A small piece of thread is attached to the specimen. Due to the evaporation of liquid nitrogen, the weight indicated by the balance shows a steady decline. A few measurements at regular intervals are made for establishing the rate of loss. The metallic piece is removed from the pan of the balance and gently lowered into the cup of liquid nitrogen by using the thread. A second set of weight measurements are made to determine the new rate of loss. From these measurements one can determine $\Delta m$, the mass of the liquid nitrogen that boiled off in cooling the piece of metal from an initial temperature $T_0$ to a final temperature $T_f=77$ K. 

Let $L$ be the latent heat of vapourization of nitrogen; $\Delta Q$, the heat gained by the liquid nitrogen, will then equal  $L\Delta m$. You have already been told how you can calculate the heat lost by the metallic specimen (see above). Use energy balance now to deduce the value of $\Theta_{\rm E}$ for Al. Since $\Delta Q$ can be equated to the change in the internal energy of the metal, we can also apply Einstein's model and write
\begin{equation}\label{eq:EinTemp}
\Delta Q =3nR[T_0{\cal E}(\Theta_{\rm E}/T_0)-T_f{\cal E}(\Theta_{\rm E}/T_f)],
\end{equation}
where $n$ is the number of moles in the specimen under test and 
\begin{equation}
{\cal E}(y)=\frac{y}{\e^y - 1}.
\end{equation}

To make matters a little concrete, take a look at Figure 6, where some real data have been plotted. The material was Al, with $n=0.21$ M, and the plot implies $\Delta m = 3.76$ g. You may take  $T_0 = 290$ K.
\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Figure6.PNG
  \includegraphics[width=12.1cm,height=9cm,keepaspectratio]{Figure6}
  \caption{\small A plot showing the rate of evaporation of liquid nitrogen before (left half) and after (right-half) the immersion of the metallic speciments in the container.}
  \label{fig:Figure6}
\end{figure}

Prepare an Excel worksheet for tabulating and analyzing the above data according to Eq.~(\ref{eq:EinTemp}).\footnote{The same worksheet will be used for analyzing the data collected by you in the laboratory.} To complete the calculation, you will need the accepted value of $L$, a task that is left for you. You may begin by setting $L=2\times 10^5$ J/kg.  Find $\mathit{\Theta}_{\rm E}$ now. Does it agree with that deduced earlier by fitting the data of G\&amp;M to Eq.~(\ref{eq:EinCv})? What is the effect of changing the value of $L$ to, say, $1.98\times 10^5$ J/kg?  What is the effect of changing the value of $T_0$ to, say, 292 K?  Develop, on the basis of such probing, a feeling for the sources of error in your experiment.

\section{In the laboratory}
\subsection{Laboratory etiquette}
You are expected to adhere to the following code:---
\begin{itemize}
  \item Do not bring any items of food or drink to the laboratory.
  \item Follow the safety precautions (for handling liquid nitrogen) that will be explained to you before you start. You should know that, under normal conditions, liquid nitrogen has a boiling point of 77 K. The experiment is inherently safe, but you should be aware that prolonged contact of the skin with liquid nitrogen will lead to severe frostbite. Care should also be exercised in handling objects that have just been taken out of a reservoir of liquid nitrogen.
  \item You must attend the lab section for which you are registered.
  \item Pre-lab assignments should be completed before you enter the room. 
  \item If you do not hand in a prelab, or if you turn in a blank prelab you will not be permitted to perform the lab. 
  \item No one will be admitted to the laboratory 10 minutes after the designated start. 
  \item Each student must have his/her own lab notebook. The pages of the notebook must be numbered in pen all the way to the end. This is to ensure that none are removed during the semester. 
  \item Labs must be written in pen, not pencil. Do not erase; put a single line through unwanted data. Neatness counts. 
  \item Students will work in pairs, but each member of the pair must record her/his own data.
  \item You will leave your work table as you found it---neat and tidy.
\end{itemize}

\subsection{Collection and analysis of data}

Carry out the measurements by following the procedure outlined above. Plot your data to see that they fit the expected pattern; if they do not, you will have to repeat the experiment. 

Prepare, using Norwegian or English, a report of your work. Give it a suitable title, and divide it into some sections. The following list is meant to serve as a guideline for you. 
\begin{description}
   \item {\bfsf \qquad Principle and task}
   \item {\bfsf \qquad Equipment}
   \item {\bfsf \qquad Set-up and procedure}
   \item {\bfsf \qquad Results \&amp; Analysis}  
   \item {\bfsf \qquad Discussion} 
\end{description}

You should hand in your report within two weeks after the completion of your experiment.


\newpage
\section{Specific heat data for aluminium}

\vspace*{1truecm}

\centerline{\includegraphics[bb=0 0 412 728,width=11.8cm,height=20.9cm,keepaspectratio]{Table2.png}}

\newpage

\vspace*{1truecm}

\centerline{\includegraphics[bb=0 0 412 626,width=11.8cm,height=17.9cm,keepaspectratio]{Table3.tif}}

\newpage

\section{Specific heat data for fluorine}

\vspace*{1truecm}

\centerline{\includegraphics[bb=0 0 412 458,width=11.8cm,height=13.1cm,keepaspectratio]{Table4.tif}}
%\renewcommand{\pagestyle}{empty}
\newpage
\section{Operating the balance}

Weighing, one of the most common tasks in the laboratory, has been greatly simplified by technological advances. This does not mean that the operator should not exercise utmost care to prevent disturbing influences. These usually involve physical changes in the sample (e.g. through slow evaporation, moisture uptake), or forces which act on the weighing pan and weighing sample (e.g. magnetism, electrostatics) and which are interpreted by the balance as weight changes.

Align the balance by checking that the air bubble is in the center of the level indicator; if it is not, use the leveling feet to make adjustments. 

\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/leveling.jpg
  \includegraphics[width=3cm,height=3cm,keepaspectratio]{leveling.jpg}
  \caption{\small The level indicator}
  \label{fig:leveling}
\end{figure}


\vspace*{1truecm}
\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Balance131.TIF
  \includegraphics[bb=0 0 901 635,width=16cm,height=11.3cm,keepaspectratio]{Balance131c.tif}
  \caption{\small Instructions for On/Off switching and simple weighing}
  \label{fig:Balance131c}
\end{figure}



%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Balance131.TIF
%  \includegraphics[bb=0 0 901 635,width=12.8cm,height=11cm,keepaspectratio]{Balance131}
%  \caption{\small Instructions for On/Off switching and simple weighing}
%  \label{fig:Balance131}
%\end{figure}
\newpage

What is meant by `taring the balance'?. Dictionary meaning of the noun {\it tare\/}: `A counterweight used in chemical analysis to balance the weight of a container'. The noun infinitive `to tare' means to `to ascertain, note, or allow for the tare of'. In a chemistry laboratory, one often weighs  a container first, and brings the weight reading to zero; this adjustment is known as taring the balance.


\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
  \centering
  % file name: C:/MyFiles/Booklet/Balance132.TIF
  \includegraphics[bb=0 0 901 635,width=16cm,height=11.3cm,keepaspectratio]{Balance132c.tif}
  \caption{\small Instructions for taring.}
  \label{fig:Balance132c}
\end{figure}


%
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Balance132.TIF
%  \includegraphics[bb=0 0 901 635,width=12.8cm,height=11cm,keepaspectratio]{Balance132}
%  \caption{\small Instructions for taring. Dictionary meaning of the noun {\it tare\/}: `A counterweight used in chemical analysis to balance the weight of a container'. The verb `tare' means to `to ascertain, note, or allow for the tare of'.}
%  \label{fig:Balance132}
%\end{figure}

%
%\begin{figure}[h] % float placement: (h)ere, page (t)op, page (b)ottom, other (p)age
%  \centering
%  % file name: C:/MyFiles/Booklet/Balance133.TIF
%  \includegraphics[bb=0 0 901 635,width=12.8cm,height=11cm,keepaspectratio]{Balance133}
%  \caption{\small }
%  \label{fig:Balance133}
%\end{figure}

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